In split signature, the twistor construction for ASD conformal structures was done by LeBrun and Mason in [1]},
and it also involves, in a way, the consideration of the space of almost-complex structures over \(M\) .
In section REF  we restricted to real twistor planes (recall that a twistor plane is the tangent space to
a twistor surface), but we can also
consider complex twistor planes by simply complexifying the space of spinors, i.e. taking
\(\mathbb {R}^{2}\otimes \mathbb {C}=\mathbb {C}^{2}\) .
Then the bundle of complex twistor planes over \(M\) , which we denote by \(\mathcal {Z}\)  following [1]},
has fibers \(\mathbb {CP}^{1}\)  (the twistor planes we consider here are the \(\beta \) -planes of [1]}).
The subbundle of real twistor planes is denoted by \(F\)  in [1]}, and its fibers are \(\mathbb {RP}^{1}\) .
Now, as we saw in Remark REF , the bundle of almost-complex structures over \(M\) , say \(\mathcal {H}\) ,
has fibers \(\mathbb {CP}^{1}\backslash \mathbb {RP}^{1}\) . This is the same as removing real twistor planes in \(\mathcal {Z}\) ,
so it coincides with the bundle \(\mathcal {Z}\backslash F\)  in [1]}.
Using that \(\mathbb {CP}^{1}\backslash \mathbb {RP}^{1}\)  is a hyperboloid of two sheets,
and identifying each sheet with a disk,
we see that \(\mathcal {H}\)  consists of two connected components, say \(\mathcal {H} = U_{+} \cup U_{-}\) ,
each of which is a disk bundle over \(M\) .
The twistor space of \(M\)  constructed in [1]}
is obtained essentially by giving a complex structure to the manifold-with-boundary \(U_{+}\cup F\) 
(see [1]}),
which we can see as “one half” (with a boundary attached) of the space of almost-complex structures over \(M\) .
A twistor space construction involving the two halves of the hyperboloid \(H^{2}\)  together also with the equator \(\mathbb {RP}^{1}\) 
(in other words, the Riemann sphere \(\mathbb {CP}^{1}=H^{2}\cup \mathbb {RP}^{1}\) )
can be found in [8]}.
Here, the twistor space is obtained in a similar way to the Atiyah-Hitchin-Singer construction [9]},
i.e. by finding an involutive subbundle for \(T\mathbb {PS}\)Whether this actually gives a complex
structure to \(\mathbb {PS}\)  in the ordinary sense is a little subtle, see [10]}.;
and it consists of two open sets that are separated by a 3-dimensional real boundary. This boundary
is the real twistor space that we considered in section REF .
